The Stacks project

Lemma 67.23.3. Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$. Then $X_{\text{ft-pts}} \subset |X|$ is the set of closed points.

Proof. If $x \in |X|$ is closed, then we can represent $x$ by a closed immersion $\mathop{\mathrm{Spec}}(k) \to X$, see Lemma 67.14.6. Hence $x$ is certainly a finite type point.

Conversely, let $x \in |X|$ be a finite type point. We know that $x$ can be represented by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field (Definition 67.6.1). On the other hand, by definition, there exists a morphism $\mathop{\mathrm{Spec}}(k') \to X$ which is locally of finite type and represents $x$ (Morphisms, Definition 29.16.3). We obtain a factorization $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \to X$. Let $U \to X$ be any étale morphism with $U$ affine and consider the morphisms

\[ \mathop{\mathrm{Spec}}(k') \times _ X U \to \mathop{\mathrm{Spec}}(k) \times _ X U \to U \]

The quasi-compact scheme $\mathop{\mathrm{Spec}}(k) \times _ X U$ is étale over $\mathop{\mathrm{Spec}}(k)$ hence is a finite disjoint union of spectra of fields (Remark 67.4.1). Moreover, the first morphism is surjective and locally of finite type (Morphisms, Lemma 29.15.8) hence surjective on finite type points (Morphisms, Lemma 29.16.6) and the composition (which is locally of finite type) sends finite type points to closed points as $U$ is Jacobson (Morphisms, Lemma 29.16.8). Thus the image of $\mathop{\mathrm{Spec}}(k) \times _ X U \to U$ is a finite set of closed points hence closed. Since this is true for every affine $U$ and étale morphism $U \to X$, we conclude that $x \in |X|$ is closed. $\square$

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